This really should have been community wiki
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Harry GindiDec 12 '09 at 5:50

I agree. I'm converting it to wiki now. In general, any question that asks for a list of general resources should be community wiki since it's not really appropriate for people to earn reputation from them (the resources should really be getting the reputation, not the people listing the resources). If you want to join the discussion on what kinds of questions should be community wiki, you can do so here: tea.mathoverflow.net/discussion/6
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Anton GeraschenkoDec 12 '09 at 6:32

A classic, but perhaps not as "geometric" as contemporary sources, is Lyndon and Schupp's Combinatorial Group Theory (named after the classic Combinatorial Group Theory, by Magnus, Karrass, and Solitar).

Lyndon and Schupp is really the only textbook source for a lot of classical material on free groups. And Magnus-Karrass-Solitar is the only textbook source for the "Magnus representation" of a free group, and thus for the residual nilpotence of free groups. However, I think both sources are hard to read for someone whose mind works in geometric ways (that might be why I secretly like them a lot!)
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Andy PutmanNov 2 '09 at 23:49

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Also, Magnus-Karrass-Solitar might be the only book ever written which is hard to read because the authors included too many details in their proofs!
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Andy PutmanNov 2 '09 at 23:50

I feel obligated to mention that if anyone ever needs to learn about Whitehead's algorithm, it's better to read Whitehead's original paper (or Stalling's "Whitehead graphs on handlebodies") than to read Lyndon and Schupp's account. Sometimes they took really topological things and made them too algebraic.
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Richard KentNov 3 '09 at 3:11

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You can even give an easy example of ascending unions of free groups not being free -- Q is an ascending union of groups each of which is isomorphic to Z! Another warning one should make about Lyndon and Schupp is that there are an astounding number of errors/misprints.
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Andy PutmanNov 3 '09 at 4:41

Pierre de la Harpe's "Topics in Geometric Group Theory" is, to be fair, the only book I know relatively well so I can't compare it to others. Anyway, I do like it - the writing style is pleasant and it gets to some non-trivial results, including a fairly complete review of the Grigorchuk group.

A book I quite like is Bogopolski's Introduction to Group Theory. It's not really an introduction (at least at undergraduate level), but it covers some things that aren't covered in the books above, particularly automorphisms of free groups and it has more Bass-Serre theory than anything I've read that's mentioned in the other answers.

I also want to add a dissenting opinion on de la Harpe's book. I think it's quite disappointing, given that it's the first real textbook since geometric group theory went beyond combinatorial group theory.

There is a very nice book related to the topic - "Word processing in groups" by David Epstein. It covers some stuff about the combinatorial aspects of geometric group theory, e.g. automatic groups, combable groups etc.

That book (according to google) is actually by Epstein, Paterson, Camon, Holt, Levy, and Thurston. It's usually referred to as Epstein, et. al.
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Peter SamuelsonJun 29 '10 at 5:03

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The title page says "Epstein, with Cannon, Holt, Levy, Paterson Thurston". The cover juxtaposes the names, but with a bigger "Epstein" and then smaller "Cannon, Holt", etc. So "Epstein et alii" doesn't seem a bad way to reference this book, but I guess one could write "Epstein cum aliis" instead :)
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Maxime BourriganApr 14 '11 at 9:08

My personal favorite learner (not a reference) for geometric group theory is John Stalling's notes from a course that he gave at Berkeley about a decade ago. It's terse, since they are just lecture notes, but I like his style of exposition and there are many examples to work through in the exercises, which I found helpful.

It's not an introductory text, but if you're trying to get a feel for the area you could look at the GGT Open Problems Wiki. It's still rather incomplete and patchy; a more coherent and shorter alternative is Bestvina's Problem List.